Problem: A composite function can be written as $w\bigl(u(x)\bigr)$, where $u$ and $w$ are basic functions. Is $g(x)=(2x+1)^4$ a composite function? If so, what are the "inner" and "outer" functions? Choose 1 answer: Choose 1 answer: (Choice A) A $g$ is composite. The "inner" function is $x^4$ and the "outer" function is $2x+1$. (Choice B, Checked) B $g$ is composite. The "inner" function is $2x+1$ and the "outer" function is $x^4$. (Choice C) C $g$ is not a composite function.
Solution: Composite and combined functions A composite function is where we make the output from one function, in this case $u$, the input for another function, in this case $w$. We can also combine functions using arithmetic operations, but such a combination is not considered a composite function. The inner function The inner function is the part we evaluate first. Frequently, we can identify the correct expression because it will appear within a grouping symbol one or more times in our composed function. Here, we have $2x+1$ inside parentheses. We evaluate this polynomial first, so $u(x)=2x+1$ is the inner function. The outer function Then we raise the entire output of $u$ to the power of $4$. So $w(x)=x^4$ is the outer function. Answer $g$ is composite. The "inner" function is $2x+1$ and the "outer" function is $x^4$. Note that there are other valid ways to decompose $g$, especially into more complicated functions.